Potential Energy Due to Continuous Masses

Potential energy is a form of energy that is stored in an object due to its position or configuration. When dealing with continuous masses, such as rods, spheres, or any object with a non-negligible size, the calculation of potential energy becomes more complex than for point masses. The potential energy in a gravitational field due to continuous masses is derived from the integration of potential energy contributions from infinitesimal mass elements.

Gravitational Potential Energy

The gravitational potential energy (U) between two point masses (m1 and m2) separated by a distance (r) is given by:

$$ U = -\frac{G m_1 m_2}{r} $$

where G is the gravitational constant.

For continuous masses, we must consider the distribution of mass throughout the object. The potential energy is then the sum of the potential energies of all infinitesimal mass elements.

Calculating Potential Energy for Continuous Masses

To calculate the potential energy of a continuous mass distribution, we divide the object into infinitesimal elements of mass (dm) and sum their contributions. This process involves integration over the volume of the object.

General Formula

The general formula for the potential energy due to a continuous mass distribution in a gravitational field is:

$$ U = -G \int \frac{dm \cdot M}{r} $$

where:

• ( dm ) is the infinitesimal mass element,
• ( M ) is the mass causing the gravitational field,
• ( r ) is the distance between the mass element and the point where the potential is being calculated.

Example: Uniform Rod

Consider a uniform rod of length ( L ) and total mass ( M ) lying along the x-axis from ( x = 0 ) to ( x = L ). To find the gravitational potential energy of a point mass ( m ) located at a distance ( R ) from one end of the rod along the x-axis, we would use the following steps:

1. Divide the rod into infinitesimal elements ( dx ).
2. Each element has a mass ( dm = \frac{M}{L} dx ).
3. The distance from the point mass to each element is ( r = R + x ).
4. Integrate over the length of the rod to find the total potential energy.

The potential energy is then:

$$ U = -Gm \int_{0}^{L} \frac{dm}{r} = -Gm \int_{0}^{L} \frac{\frac{M}{L} dx}{R + x} $$

After integration, we get:

$$ U = -GmM \ln\left(\frac{R + L}{R}\right) $$

Differences and Important Points

Aspect	Point Masses	Continuous Masses
Basic Concept	Potential energy is calculated for discrete points.	Potential energy is calculated for a distribution of mass.
Calculation	Direct use of the formula with given masses and distance.	Integration over the mass distribution is required.
Mass Element	Treated as a single point with mass ( m ).	Infinitesimal elements ( dm ) are considered.
Distance	Distance ( r ) is fixed between two points.	Distance ( r ) varies and is part of the integration.
Examples	Gravitational potential energy between Earth and a satellite.	Gravitational potential energy of a point mass outside a spherical shell.

Conclusion

Understanding potential energy due to continuous masses is crucial for solving problems in gravitational physics, especially when dealing with extended bodies. The key is to break down the mass distribution into infinitesimal elements and integrate their potential energy contributions. This approach allows for the calculation of potential energy in various configurations and is essential for students preparing for exams in physics.